Start with this thesis:
- A random belief of a random person is at least as likely true as not.
Now observe this:
- A random atomic proposition is significantly more likely false than true.
I now claim:
- A random proposition is significantly more likely to be false than true.
The method is this. I shall suppose that the basic connectives are "and", "or" and "not", and that we have a stock of basic predicates and names for all objects, one name per object. We first randomly choose an item from the set of basic connectives and basic predicates. If the item is an n-ary predicate P, we randomly choose a sequence (n1,...,nn) of n names, and write down the sentence P(n1,...,nn). If the item is a unary connective (i.e., "not"), we write down the connective followed by a random sentence (we recurse here). If the item is a binary connective, we write down a random sentence (recursing) in parentheses, followed by the connective, followed by another random sentence (recursing again) in parentheses. The recursion is not logically guaranteed to finish, but we can conditionalize on the recursion finishing (plus I think it's going to finish with probability one if there are enough predicates).
Now, let p0 be the probability that a random atomic sentence is true. Let N be the number of predicates. Let p be the probability that a sentence generated by the above procedure is true. Then:
Now an interesting result follows from (1)-(3). The seemingly innocuous claim (1) commits us to assigning a pretty substantive amount of weight to a Principle of Credulity. For the fact that someone believes a proposition raises the probability from something that according to (3) was significantly less than 1/2 at least to 1/2. Thus the mere fact that someone believes something is significant evidence for its truth, even if it does not suffice for making the conclusion reasonable to believe.
In fact, I suspect that p0 is very small, maybe as small as 1/100 or even much smaller. In that case, the probability of a random proposition being true might be very small. And yet we have (1). So belief is significant evidence.